800 mg of a medication is administered to a patient. After 6 hours, only 219 mg remains in the bloodstream. If the decay is continuous, what is the continuous decay rate (as a percentage)?

To find the continuous decay rate, we can use the formula for exponential decay: \[ A(t) = A_0 \times e^{-kt} \] where: - \( A(t) \) is the amount of medication remaining at time \( t \), - \( A_0 \) is the initial amount of medication, - \( k \) is the decay rate, - \( t \) is the time. Given: - Initial amount of medication, \( A_0 = 800 \) mg, - Amount of medication remaining after 6 hours, \( A(6) = 219 \) mg, - Time, \( t = 6 \) hours. We need to find the decay rate, \( k \). Substitute the given values into the formula: \[ 219 = 800 \times e^{-6k} \] Now, we can solve for \( k \) using the natural logarithm: \[ \ln\left(\frac{219}{800}\right) = -6k \] \[ k = -\frac{1}{6} \times \ln\left(\frac{219}{800}\right) \] Calculate the value of \( k \) and then convert it to a percentage. The continuous decay rate is approximately 0.215923. To convert this to a percentage, we multiply by 100: \[ 0.215923 \times 100 \approx 21.5923\% \] Therefore, the continuous decay rate is approximately 21.59%.